\appendix
%\numberwithin{equation}{section}

\begin{center}
  {\bf Appendix A} \\  
  {\bf Analytical solution of the decay chain} 
\end{center}

The simplified decay chain system (Fig.\ref{fig:decay_chain}b) 
can be described by an ordinary differential
equation system with four unknowns (in activity concentration form):  
\begin{eqnarray}
  && \frac{dc_1}{dt} = -{\lambda_1}c_1,
\label{eqn:a1}\\
  && \frac{dc_2}{dt} = \frac{\lambda_1}{\lambda_2}c_1 - {\lambda_2}c_2,
\label{eqn:a2}\\
  && \frac{dc_3}{dt} = \frac{\lambda_2}{\lambda_3}c_2 - {\lambda_3}c_3,
\label{eqn:a3}\\
  && \frac{dc_4}{dt} = \frac{\lambda_3}{\lambda_4}c_3,
\label{eqn:a4}
\end{eqnarray}
where $c_1$, $c_2$, $c_3$, and $c_4$ are the activity concentration of
$^{222}$Rn, $^{214}$Pb, $^{214}$Bi, and $^{210}$Pb, respectively, and
$\lambda_1$, $\lambda_2$, $\lambda_3$, and $\lambda_4$ are the corresponding
decay constants. 
For each model time step ($\Delta t = 12$~min), the analytical solution of
the decay chain at $t+\Delta t$ reads
\begin{eqnarray}
  && c_1(t+\Delta t) = c_1(t)\,e^{-\lambda_1\Delta t}\,,
\label{eqn:c1}\\
  && c_2(t+\Delta t) = c_2(t)\,e^{-\lambda_2\Delta t}
% && \quad\quad\quad\quad\quad
%    + \frac{\lambda_2\,c_1(t)}{\lambda_2-\lambda_1}
%      \left(e^{-\lambda_1\Delta t}-e^{-\lambda_2\Delta t}\right)\,,
     +\,\chi_{21}\,\eta_{12}\,c_1(t)\,,
\label{eqn:c2}\\
  && c_3(t+\Delta t) = c_3(t)\,e^{-\lambda_3\Delta t}
% && \quad\quad\quad
%    + \frac{\lambda_2\,\lambda_3\,c_1(t)}{(\lambda_3-\lambda_1)(\lambda_2-\lambda_1)}
%      \left(e^{-\lambda_1\Delta t}-e^{-\lambda_3\Delta t}\right) \nonumber \\
     +\,\chi_{21}\,\chi_{31}\,\eta_{13}\,c_1(t)  
\label{eqn:c3}\\
% && + \frac{\lambda_3}{\lambda_3-\lambda_2}
%      \left(c_2(t)-\frac{\lambda_2\,c_1(t)}{(\lambda_2-\lambda_1)}\right)
%      \left(e^{-\lambda_2\Delta t}-e^{-\lambda_3\Delta t}\right)\!.
  && \quad\quad\quad\quad\quad
     +\,\chi_{32}\,\eta_{23}\,\left(c_2(t)-\chi_{21}\,c_1(t)\right)\!. \nonumber 
\end{eqnarray}
where 
\begin{eqnarray}
 && \chi_{21} = \frac{\lambda_2}{\lambda_2-\lambda_1}\,, 
    \chi_{31} = \frac{\lambda_3}{\lambda_3-\lambda_1}\,, 
    \chi_{32} = \frac{\lambda_3}{\lambda_3-\lambda_2}\,, \nonumber \\
 && \eta_{12} = \left(e^{-\lambda_1\Delta t}-e^{-\lambda_2\Delta t}\right)\,,\nonumber \\ 
 && \eta_{13} = \left(e^{-\lambda_1\Delta t}-e^{-\lambda_3\Delta t}\right)\,,\nonumber \\ 
 && \eta_{23} = \left(e^{-\lambda_2\Delta t}-e^{-\lambda_3\Delta t}\right)\!.\nonumber   
\end{eqnarray}
By integrating Eqns.(\ref{eqn:c1})--(\ref{eqn:c3}) from t to $t+\Delta t$, 
the time-step average concentration can be obtained: 
\begin{eqnarray}
 && \bar{c}_1 = \theta_{1}\,c_1(t)\,,
\label{eqn:d1}\\
 && \bar{c}_2 = \theta_{2}\,c_2(t)
                +\,           \chi_{21}\,\left(\theta_{1}-\theta_{2}\right)\,c_1(t)\,,
\label{eqn:d2}\\
 && \bar{c}_3 = \theta_{3}\,c_3(t) 
                +\,\chi_{21}\,\chi_{31}\,\left(\theta_{1}-\theta_{3}\right)\,c_1(t) 
\label{eqn:d3}\\
 && \quad\quad\quad\quad\quad\quad 
                +\,\chi_{32}\,\left(\theta_{2}-\theta_{3}\right)\,\left(c_2(t)-\chi_{21}\,c_1(t)\right) \!. \nonumber
\end{eqnarray}
where
\begin{eqnarray}
 && \theta_{1} = \frac{\lambda_1-e^{-\lambda_1\Delta t}}{\lambda_1\,\Delta t}\,, 
    \theta_{2} = \frac{\lambda_2-e^{-\lambda_2\Delta t}}{\lambda_2\,\Delta t}\,, \nonumber \\
 && \theta_{3} = \frac{\lambda_3-e^{-\lambda_3\Delta t}}{\lambda_3\,\Delta t}\!. \nonumber 
\label{eqn:e1}
\end{eqnarray}

Since the decay of $^{210}$Pb is ignored, its concentration is not computed in the model. 

